ECE600 F13 set theory review mhossain - Rhea

Random Variables and Signals

Subtopic 1: Set Theory Review

Definitions and Notation

Definition $\qquad$ A set is a collection of objects called elements, numbers or points.

Notation $\qquad$ $\omega$ ∈ $$ A $$ means $\omega$ is an element of the set $$ A $$ . $\omega$ ∉ $$ A $$ means $\omega$ is not in the set $$ A $$ .

There are two common ways to specify a set:

Explicitly list all elements, e.g. $\{1,2,3,4,5,6\}$
Specify a rule for membership, e.g. $A=\{\omega$ ∈ $$ Z:1 $$ ≤ $\omega$ ≤ $6\}$ , i.e. the set of all integers between 1 and 6 inclusive. I prefer this notation for large or infinite sets.

Note that there is always a set that contains every possible element of interest. This set, along with some structure imposed upon the set, is called a space, denoted

\mathcal{S}

\qquad

or

\qquad

\Omega .\

Definition $\qquad$ Let $$ A $$ and $$ B $$ be two sets. Then,

\begin{align} A = B &\Longleftrightarrow \omega \in A \Rightarrow \omega \in B \;\and\; \omega \in B \Rightarrow \omega \in A \\ &\Longleftrightarrow A \subset B \;\and\; B \subset A \end{align}

The proof that the second statement is equivalent is trivial and follows from the definition of the term subset, which is presented shortly.

If $$ A $$ and $$ B $$ are not equal, we write

A\neq B

Definition $\qquad$ If $\omega$ ∈ $$ A $$ ⇒ $\omega$ ∈ $$ B $$ , then $$ A $$ is said to be a subset of $$ B $$ . if If $$ B $$ contains atleast one element that is not in $$ A $$ , then $$ A $$ is a proper subset of $$ B $$ . We will simply call $$ A $$ a subset of $$ B $$ in either case, and write $$ A $$ ⊂ $$ B $$ .

Definition $\qquad$ the set with not elements is called the empty set, or null set and is denoted Ø or {}.

Venn Diagrams

A Venn diagram is a graphical representation of sets. It can be useful to gain insight into a problem, but cannot be used as a proof.

Fig 1: An example of a Venn diagram

Set Operations

Definition $\qquad$ The intersection of sets $$ A $$ and is defined as

A\cap B \equiv \{\omega \in \mathcal{S}: \omega \in A \and \omega \in B\}

Fig 2: A∩B is shown in green

Definition $\qquad$ The union of sets $$ A $$ and is defined as

A\cup B \equiv \{\omega \in \mathcal{S}: \omega \in A \or \omega \in B \; or\; both\}

Fig 3: A∪B is shown in green

Definition $\qquad$ The complement of a set $$ A $$ , denoted $$ A^c $$ , Ā, or A' is defined as

A^c \equiv \{\omega \in \mathcal{S}: \omega \notin A\}

Fig 4:

$ A^c $

is shown in green

Definition $\qquad$ The set difference $$ A-B $$ or A\B is defined as

A-B \equiv \{\omega \in \mathcal{S}: \omega \in A\and \omega \notin B\}

Note that

A-B=A\cap B^c

Fig 5:

$ A-B $

is shown in green

Definition $\qquad$ Sets $$ A $$ and $$ B $$ are disjoint if $$ A $$ and $$ B $$ have not elements in common ie

A\cap B = \varnothing

In figure 1, A and C are disjoint. B and C are also disjoint.

Algebra of Sets

References

M. Comer. ECE 600. Class Lecture. Random Variables and Signals. Faculty of Electrical Engineering, Purdue University. Fall 2013.